Waveguide laser

ABSTRACT

A waveguide laser (10) incorporates a guide (12) and two concave resonator mirrors (14, 16). The guide (12) is of square section with side (2a), and of length L equal to 4a 2  /λ, where λ is a laser operating wavelength. The mirrors (14, 16) are phase matched to respective Gaussian intensity profile radiation beams with beam waists at respective waveguide end apertures (20, 22). Each beam waist has a radius w 0  in the range 0.1a to 0.65a to avoid waveguide edge effects and excitation of unwanted high order waveguide modes. The laser (10) has good transverse spatial mode characteristics.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to a waveguide laser.

2. Discussion of Prior Art

Waveguide lasers are known in the part art. Such as laser typicallyconsists of two mirrors (or equivalent reflecting devices) defining anoptical resonator cavity, together with a waveguide defining at leastpart of an optical path between the reflectors. The waveguide has endapertures at or near which the reflectors are positioned respectively.The reflectors' radii of curvature and their positioning relative to thewaveguide are related by the following Equations (1) and (2):

    R=z.sub.m +B.sup.2 /z.sub.m                                ( 1)

    w=w.sub.0  1+(z.sup.2 /B.sup.2)!.sup.1/2                   ( 2)

where:

R is the radius of curvature of the respective mirror in each case,

z is a position coordinate measured along the laser beam from eachmirror to the respective nearest waveguide end aperture,

z_(m) is the value of z at the respective mirror,

B is the confocal beam parameter equal to πw₀ ² /λ,

w is the beam radius at position z, and is measured between positions atwhich beam intensity is maximum and 1/e² of maximum,

w₀ is the beam waist radius of a TEM₀₀ intensity profile laser beammeasured at the respective neighbouring waveguide and aperture, and

λ is the wavelength of laser radiation measured in the respective regionbetween mirror and waveguide.

Equations (1) and (2) define a situation in which a mirror of radius Ris phase matched to a TEM₀₀ beam. Waveguide laser resonators haveassociated mirror configurations referred to in the prior art as Case I,Case II and Case III. They are defined with reference to Equations (1)and (2) above. They are described by J. J. Degnan and D. R. Hall, III, JQuantum Electron, Vol QE-9, pp 901-910, 1973. They are also referred toin "Theory of Waveguide Laser Resonators", Chapter 3 of "The physics andTechnology of Laser Resonators", edited by D. R. Hall and P. E. Jackson,published by Adam Hilger. A Case I mirror has a large radius ofcurvature R (possibly infinite, ie a plane mirror) and a small or zerovalue of z; ie R tend to B² /z in Equation (1) as z goes to zero. A CaseII reflector has a large radius of curvature and is positioned such thatz is approximately equal to R, B² /z being negligible. Finally, a CaseIII reflector is one with z equal to about half the value of R, z beingapproximately equal to B and w₀ being chosen to provide optimum couplingto the EH₁₁ fundamental waveguide mode.

Waveguide lasers incorporating gas media are advantageous because thewaveguide provides cooling for the discharge. As a result of gasdischarge scaling laws, the waveguide also allows high pressureoperation. Moreover, CO₂ lasers in particular have a laser line widththat increases with increased operating pressure, so incorporation of awaveguide improved potential tuning range. This also applies to othergas lasers in which laser line width increases with increasing pressure.A further potential advantage is that the gain medium of a waveguidelaser may be confined to a small dimension optical waveguide, whichmakes it very compact compared with a free space resonator. Moreover,the resonator mode may effectively fill the waveguide, producing goodoverlap between the optical field and the gain medium. This results inefficient extraction of optical power. It is not necessarily the case infree space resonator designs.

However, waveguide lasers suffer from the disadvantage that thewaveguides are difficult to fabricate with sufficient accuracy to obtainacceptable laser performance. A typical CO₂ laser has an alumina (Al₂O₃) waveguide in the region of 30 cm in length with an internal bore ofsquare cross-section of side 2 mm. It is very difficult to fabricate aninternal bore of these small dimensions accurately over the whole lengthof the waveguide. Uncertainty of cross-section leads to uncertainty oflaser transverse mode characteristics. Waveguide lasers also suffer fromthe major disadvantage that they tend to run on unwanted higher orderwaveguide resonator modes rather than the fundamental resonator mode(usually near TEM₀₀). This is particularly true for Case I designs. CaseIII is better in this respect, but it has an added disadvantage that itrequires a concave mirror placed a much longer distance from thewaveguide. Consequently, there is reduced effective power output perunit length of the laser compared to Case I.

In "Radio Frequency Excited CO₂ Waveguide Lasers", Rev Sci Instrum 55(1984), pp 1539-1541, R. L. Sinclair and J. Tulip described waveguideresonators based on square cross-section waveguides with twoapproximately Case I reflectors. The waveguide consists of two sectionseach 29.5 cm long with cross-sections of either 2.0 mm or 2.5 mm side.The waveguides are defined by walls, these being formed of aluminium onone side and alumina on the remaining three sides. The reflectors areeach zinc selenide coated and positioned 2.0 mm from the waveguideapertures. This laser suffers from relatively poor transverse modediscrimination, and the mode quality of the laser output is easilydegraded by perturbations in the laser discharge or in the laser optics.

A different form of waveguide laser is described by J-L. Boulnois and G.P. Agrawal in "Mode Discrimination and Coupling Losses inRectangular-Waveguide Resonators with Conventional and Phase-ConjugateMirrors", J Opt Soc Am 72 (1982), pp 853-860. This incorporated analumina waveguide of length 200 mm and square section with side 2 mm.Curved mirrors each with radius of curvature R equal to 1000 mm werepositioned 21 mm from respective end apertures of the waveguide. Themirrors did not conform to Case I, II or III. They were phase matched toa free space TEM₀₀ mode within the laser resonator cavity but outsidethe waveguide. Phase matching maximised the excitation of the waveguidefundamental mode EH₁₁ and ensured the highest possible efficiency ofradiation intensity coupling between the TEM₀₀ free space mode and theEH₁₁ waveguide mode. This type of laser is intolerant to waveguidemanufacturing errors because its design ignores multimode coupling andpropagation effects.

S. N. Chirikov, S. T. Kornilov, E. D. Protsenko and M. I. Pschikov in"Formation Details of a Waveguide Gas Laser Intensity Distribution",Infrared Phys. 30, (1990), pp 455-464, describe a laser resonator with asquare section waveguide and two distant plane mirrors. For the purposesof studying the mode phase shifts, the mirrors were however treated asequivalent to Case I. The authors investigated the effect on theresonator output of altering the length of the waveguide. In particularthey studied the contribution of different waveguide modes to theresonator modes, and the losses of the resonator modes, as the waveguidelength changed. The results showed that certain laser properties such assensitivity to mirror misalignment depend on waveguide Fresnel number N;N is defined as a² /λL, where a is waveguide half-width, L is waveguidelength and λ is radiation wavelength. Similar results were obtained byC. A. Hill, P. Monk and D. R. Hall, IEEE J. Quantum Electron, Vol. QE-23pp 1968-1973, 1987.

C. A. Hill in "Transverse Modes of Plane-Mirror Waveguide Resonators",IEEE J Quantum Electron. QE-24, (1988), pp 1936-1946, discusses thetheory of square section waveguides with plane mirrors. It is shown thatit is difficult in this kind of laser to combine low loss with good modediscrimination without sensitivity to waveguide manufacturing errors.

It is also known to employ lasers with circular bore waveguides. Theseare described by F. P. Roullard III and M. Bass, IEEE Quantum Electron,Vol. QE-13, pp 3684-3690, 1977, and by M. Lyszyk et al, Opt. Commun,Vol. 36, pp 327-330, 1981. Generally, circular bore waveguide laserssuffer from the disadvantage that laser output mode properties are notsufficiently good for high performance applications. Furthermore,circular bore waveguides are much more difficult to manufactureaccurately compared to waveguides with planar parallel walls.

It is a very important requirement of many lasers that they produce anoutput beam directed along the laser axis with a high on-axis intensityin the fare filed. This means that the output should be a fundamentalspatial mode having a single lobe of TEM₀₀ intensity profile centred onthe laser axis. It is in general difficult to achieve this reliably.Lasers may produce two or more output modes, which may be at differentfrequencies. They may also produce output modes with off-axis beamlobes. Moreover, apparently identical lasers may produce differentoutputs, and a single laser may change abruptly from single lobe tomulti-lobe during warm-up or because of change in ambient conditions. Amulti-lobe pattern is not useful for most purposes, in that most opticalsystems involving lasers are designed for the laser beams arepotentially hazardous, in that radiation is directed in non-designdirections possibly out of the optical system.

SUMMARY OF THE INVENTION

It is an object of the invention to provide an alternative form ofwaveguide laser.

The present invention provides a waveguide laser including a waveguidelocated in a laser resonator cavity defined by first and secondreflecting means, and wherein:

(a) the waveguide has at least one pair of substantially planar guidewalls which are substantially parallel to one another and separated fromone another by a distance 2a;

(b) the cavity is designed to provide a beam waist of magnitude 2w₀located centrally of a waveguide end apertures, where w₀ is in the range0.1a to 0.65a;

(c) the first reflection means is located to receive radiation emergentfrom the waveguide through the end aperture, and has converging andreflecting properties which, at least in a dimension orthogonal to theguide walls, are arranged to be phase matched to radiation received froman amplitude distribution at the aperture of TEM₀₀ form having the saidbeam waist magnitude; and

(d) the cavity is designed to be electric field preserving at thewaveguide end aperture such that a radiation amplitude distribution atthis aperture of TEM₀₀ form and having the said beam waist magnitude isdesigned to be recreated after radiation therefrom has passed throughthe waveguide to the second reflecting means and returned.

The invention provides the advantage that it is capable of providing anoutput beam which has higher TEM₀₀ mode content in the fundamental modethan the prior art. It is also in particular embodiments capable ofexhibiting greater insensitivity to manufacturing errors. It alsoprovides the advantage that undesirable waveguide input edge effects areavoided and so also is disproportionate attenuation of high order modes.

In a preferred embodiment, the beam waist radius w₀ is in the range 0.3ato 0.65a; the laser has a gain medium within the waveguide providinggain at an operating wavelength within the waveguide of λ, and thewaveguide is of square cross-section with side 2a and length 4na² /λwhere n is a positive integer, the aperture is a first end aperture andthe waveguide has a second end aperture at which the cavity is arrangedto be electric field preserving.

The waveguide may be a first waveguide and the laser may include asecond waveguide within the cavity. The laser may include means forcoupling radiation from the first waveguide to the second waveguide,which means may define mutually inclined optical paths in the first andsecond waveguides.

The laser may alternatively include second reflecting means comprising aplane mirror; the waveguide may be of square cross-section with side 2aand length 2 a² /λ, the aperture may be a first end aperture and thewaveguide may have a second end aperture arranged immediately adjacentthe second reflecting means.

In a further embodiment of the laser of the invention:

(a) the waveguide is a first waveguide,

(b) the aperture is one of two end apertures of the first waveguide,

(c) a second waveguide having two end apertures is arranged within thecavity,

(d) the laser includes means for coupling radiation between one endaperture of the first waveguide and one end aperture of the secondwaveguide,

(e) both waveguides are of square cross-section with side 2a and length4a² /λ, and

(f) the cavity is arranged to be electric field preserving at the otherend aperture of the second waveguide.

In this further embodiment, the means for coupling radiation may definemutually inclined optical paths in the first and second waveguides.

BRIEF DESCRIPTION OF THE DRAWINGS

In order that the invention might be more fully understood, embodimentsthereof will now be described, with reference to the accompany drawings,in which:

FIG. 1 is a schematic sectional side view of a waveguide laser of theinvention incorporating two like concave resonator mirrors;

FIG. 2 illustrates electric field intensity distributions within thewaveguide of the laser of FIG. 1;

FIGS. 3 & 4 are schematic sectional side views of waveguide lasers ofthe invention each incorporating a plane resonator mirror and a concaveresonator mirror;

FIG. 5 graphically illustrates TEM₀₀ transmission fidelity within awaveguide as a function of waveguide length;

FIG. 6 graphically illustrates round-trip resonator loss as a functionof waveguide length in a laser;

FIG. 7 schematically illustrates the effect of waveguide length changeon laser beam waist position;

FIG. 8 is a schematic sectional side view of a laser of the inventionincorporating two optically coupled waveguides;

FIG. 9 is a schematic sectional side view of a laser of the inventionincorporating a lens/grating combination; and

FIG. 10 is a schematic sectional side view of a laser of the inventionincorporating two waveguides optically coupled by two lenses and a planemirror.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Referring to FIG. 1, there is shown a sectional side view of a waveguidelaser of the invention indicated generally by 10. The laser 10 is notdrawn to scale. It incorporates an optical waveguide 12 (hereinafterreferred to as a "guide") positioned between first and second convergingmirrors 14 and 16. The mirrors 14 and 16 are fully reflecting andpartially reflecting respectively. The guide 12 and mirrors 14 and 16have a common optical axis 18, which is in the plane of the drawing.

The guide 12 is a hollow alumina tube with square cross-section of side2a equal to 2 mm. It is of length L given by:

    L=4a.sup.2 /λ                                       (3)

where λ is the laser operating wavelength measured within the guide 12.

The laser 10 is designed for operation at a free space wavelength of10.59×10⁻⁴ cm. The refractive index of the CO₂ medium within the guide12 is substantially equal to unity, and the wavelength λ in the guide istherefore equal to its free space value. As calculated from Equation(3), L is 37.8 cm.

The mirrors 14 and 16 have respective radii of curvature R₁ and R₂ bothequal to 29 cm in this example. They are spaced apart from respectiveguide end apertures 20 and 22 by respective distances z₁ and z₂. In thepresent example, z₁ and z₂ are both equal to 4.7 cm. The laser 10 isdesigned to produce radiation of TEM₀₀ intensity profile in planesorthogonal to the axis 18, and defined by:

    I.sub.r (z)=I.sub.0 (z)e.sup.-2r.spsp.2.sup./w.spsp.2      (4)

where:

I₄ (z) is the radiation intensity in any plane between a mirror 14 or 16and the waveguide 12 distant z along the axis 18 from the waveguide,

I₀ (z) is the radiation intensity measured at the point z on the axis18,

r is the radial distance from the axis 18 of the point at which I₄ (z)is determined, and

w is the laser beam radius at the axial position z, and is defined asthe value of r at which I₄ (z)/I₀ (z) is e⁻² ; it is a function ofdistance between the guide 12 and mirror 14 or 16.

The laser is designed to produce two minimum beam widths (referred to asbeam waists) of design radius w₀ (diameter 2w₀) at and concentric withfirst and second guide end apertures 20 and 22 respectively. Theseapertures lie in planes 24 and 28 indicated by chain lines 24 and 28 andextending perpendicular to the plane of the drawing. The design beamwaist diameter 2w₀ is related to mirror radius of curvature andmirror-guide spacing by Equation (1) repeated below for convenience:

    R=z+B.sup.2 /z, where B=πw.sub.0.sup.2 /λ        (1)

R=R₁ =R₂ in the FIG. 1 example.

Eliminating B: ##EQU1##

Substituting for R, z and λ in Equation (5) gives:

    w.sub.0 =0.6 mm                                            (6)

The laser 10 consequently is designed to produce radiation 30 with adesign beam waist radius of 0.6 mm (diameter 2w₀ of 1.2 mm) in theplanes 24 and 28. The ratio w₀ /a of design beam waist diameter towaveguide diameter is 0.6; ie w₀ is equal to 0.6 z.

The radius of curvature R of each mirror 14 or 16 is chosen so that itis accurately phase-matched to a conceptual spherical wavefront whichwould be produced at that mirror by a TEM₀₀ intensity profile planarwavefront located in a respective plane 24 or 28 and centred on an endaperture 20 or 22. To a first order approximation a TEM₀₀ Gaussianwavefront at a guide aperture 20 or 22 produces a substantiallyspherical wavefront at a mirror 14 or 16. Each mirror consequentlyproduces retroreflection of incident radiation across the incidentwavefront in each case. On receipt of such a spherical wavefront, eachmirror 14 or 16 therefore returns to the respective guide aperture 20 or22 a substantially fully phase-reversed beam which recreates the planarTEM₀₀ wavefront at that aperture. As has been said, the second mirror 16is partially reflecting, and transmission through it gives rise to anoutput beam 32.

The distance z of each mirror 14 or 16 from the respective guideaperture 20 or 22 is chosen to provide significant diffraction andconsequent laser beam divergence between the relevant aperture andmirror in each case. Such diffraction is in part responsible foradvantageous mode selection properties, involving preferentialrecreation of a TEM₀₀ mode leaving an aperture 20 or 22 and returning toit from a mirror 14 or 16. The relevant TEM₀₀ mode is that having thedesign beam waist w₀ for which the mirrors are selected in accordancewith Equation (1). Modes with beam waists not equal to w₀ are lessaccurately recreated on return after reflection from a mirror 14 or 16,which normally results in higher loss for the associated resonator mode.In accordance with the invention, R must be not greater than 5B and notless than 2B, where B is the confocal beam parameter in Equation (1).This produces acceptable diffraction properties as aforesaid combinedwith reasonable compactness of the laser. Lasers of the inventiontherefore lie outside both Case I (R>>B, z<<B) and Case II (Z and R both>>B).

As will be described later in more detail, the guide 12 is designed toreproduce at one aperture 20 or 22 any electric field amplitudedistribution input to the other guide aperture 22 or 20, provided thatthe input distribution excites only symmetric modes of the guide. Anon-axis, in phase, plane wave excites only symmetric modes. Inconsequence, as regards the form of the fundamental resonator mode(although not in other respects), the laser resonator device 10 behavessubstantially as through the guide 12 were removed and the mirrors 14and 16 were moved together until the planes 24 and 28 were coincident.Moreover, the mirrors 14 and 16 have radii of curvature and positioningdesigned to provide for a TEM₀₀ intensity profile radiation beam with abeam waist at an aperture 20 or 22 to be returned to that aperture bythe respective mirror 14 and 16 with unchanged relative transverse phaseand amplitude profiles. This is achieved by providing for mirror radiiR₁ and R₂ in each case to satisfy Equation (1) and for the mirrors 14and 16 to have centres of curvature (not shown) which are on the guideaxis 18; ie the guide 12 and mirrors 14 and 16 form a well-alignedcoaxial system. Equation (1) determines the design beam waist radius w₀for given values of R, z and λ.

The net effect of the positioning and dimensions of the guide 12 andmirrors 14 and 16 is that the laser 10 has comparatively low loss for aTEM₀₀ beam with the selected laser beam waist, but comparatively highloss otherwise. When a CO₂ laser medium within the guide 12 is excited,the resonator mode having the highest ratio of gain to loss is the lasermode that is generated. Other laser spatial modes of higher gain/lossratio are suppressed. A laser of the invention has a lowest loss modewhich is the fundamental quasi-TEM₀₀ mode.

It has been found that it is advantageous to avoid very high orderspatial modes of propagation in the guide 12 and also to avoid edgeeffects at the guide apertures 20 and 22. To achieve this, the radiationintensity at the edges of the apertures 20 and 22 should be less than 1%of maximum intensity on the axis 18 at each beam waist. This sets anupper limit on the ratio of beam waist to aperture size, is w₀ /a, of0.65. Such an upper limit provides for input radiation intensity of aTEM₀₀ beam at aperture edges to be less than 1% of the maximum on-axisbeam intensity.

If the ratio w₀ /a is greater than or equal to 0.1, but not greater than0.65, there will be insignificant excitation of very high order spatialmodes within the guide 12; ie guide modes EH_(mn) with m and n equal to11 or more will not be excited to any appreciable extent. If w₀ /a isgreater than or equal to 0.3, but not greater than 0.65, then guidemodes above EH₇₇ will have intensities less than 0.5% of the totalradiation intensity. It is advantageous to avoid significant excitationof very high order guide modes, because they suffer fromdisproportionately large attenuation in propagation along the guide 12.They interact more strongly with guide walls than lower order modes.This introduces phase errors which degrade electric field regeneration.In consequence, such very high order modes are not available in thecorrect relative proportions for reproduction at an output aperture 20or 22 of an electric field distribution initially input at aperture 22or 20 respectively. The ratio w₀ /a of the input radiation beam waist tothe input aperture size should therefore be appropriate to discriminateagainst excitation of very high order modes (m, n>11); w₀ /a shouldtherefore be in the range 0.1 to 0.65, and preferably in the range 0.3to 0.65 for modes above EH₇₇ to receive less than 1% of total radiationintensity. Devices of the invention may conveniently have w₀ /a in therange 0.4 to 0.55.

In the device 10, w₀ /a is 0.6, R₁ and R₂ are equal to 20 cm and z₁ andz₂ are equal to 4.7 cm. Equation (1) is therefore satisfied. A TEM₀₀electric field intensity distribution characterised by this waist sizeand leaving the guide 12 for either of the mirrors 14 and 16 is returnedto the guide substantially unchanged. This ignores imperfections of andedge effects at the relevant mirror, which are insignificant in practicefor a mirror of adequate diameter. Moreover, the electric fielddistribution passing into the guide 12 at one of the apertures 20 and 22is designed to be reproduced without significant change at the otheraperture, and to pass to the other mirror for retroflection and returnthrough the guide 12 as before.

Referring now also to FIG. 2, there are shown calculated plots of thetransverse electric field intensity distribution due to radiationpropagation in a single pass along the guide 12 at various points alongthe guide length L. In this drawing the longitudinal coordinate z ismeasured along the guide 12, and z values of 0 and L are at respectiveend aperture planes 24 and 28. The plots are referenced 40 to 48, andcorrespond to intervals of L/8 along the guide 12; ie plot N is thetransverse electric field intensity distribution I(x,y) at a value of zof (N-40)L/8, where N is the plot reference number in the range 40 to48.

Plot 40 shows the input excitation of TEM₀₀ form received at the firstguide aperture 20 from the first mirror 14. This input excitationbecomes decomposed into a linear combination of the modes of the guide12. As has been said, only symmetric modes are excited. The modespropagate at different rates only the guide 12; ie model dispersionoccurs. In consequence, at the mid-length of the guide 12 where z is L/2the intensity distribution is a four-lobed pattern 44 arising fromintermode interference due to phase differences having arisen betweenmodes. At the far end of the guide 12 where z is L, the symmetric modesare in phase once more and give rise to a single lobed intensitydistribution 48 equivalent to 40. The input intensity distribution 40 istherefore recreated as 48 at the second aperture 22.

In operation, radiation generated in a laser resonator arises frommultiple transits of the resonator cavity. This establishes aself-consistent electric field which repeats itself in phase andamplitude after each round trip of the cavity. The laser 10 is designedfor radiation to propagate from the second aperture 22 as a single-lobedquasi-TEM₀₀ beam to the second mirror 16, and is partially transmittedand partially reflected to provide the output beam 32. This beamtherefore consists substantially of a single lobe with maximum intensityon the laser axis 18. The radiation reflected at the second mirror 16retraces the path of the beam 30; it recreates the intensitydistributions 48 to 40 in reverse order within the guide 12 beforereaching the first mirror 14 and returning once more. It is an importantadvantage of the invention that the output beam 32 arises from aquasi-TEM₀₀ mode, since this provides a single lobed radiation beam withmaximum intensity on the device axis 18. Higher order resonator modesproduce off-axis lobes which are less useful for most optical purposesand may be hazardous.

The field-reproducing properties of the square section guide 12 arisefrom the general propagation characteristics of a rectangular waveguide.This latter waveguide is taken to have height 2a and width 2b, and to bebounded by a homogeneous dielectric material with complex dielectricconstant .di-elect cons.. It is also assumed that this dielectricmaterial (which provides the waveguide walls) is highly reflecting andnot significantly attenuating for required propagating modes. Thewaveguide has height, width and length dimensions which are parallel tothe x, y and z axes respectively. It has normalised linearly polarizedmodes of the kind EH_(mn). The electric field contribution E_(mn)(x,y,z) of the mnth mode EH_(mn) at the point (x,y,z) has beencalculated by Laakmann et al in Appl. Opt. Vol. 15, No 5, pages1334-1340, May 1976 as follows: ##EQU2## where m is the mode numberrelating to the field dependency along the x axis,

n is the mode number relating to the field dependence along the y axis,

z is the distance along the z axis (equivalent to axis 18 in FIG. 1),

γ_(mn) =(β_(mn) +iα_(mn)), the propagation constant of the mn^(th) mode,β_(mn) and α_(mn) being the mn^(th) mode's phase and attenuationcoefficients, and

"cos" above "sin" indicates the former applies to odd mode numbers (n orn as appropriate) and the latter to even mode numbers.

The phase coefficient β_(mn) is given by: ##EQU3##

If the negative term in parenthesis in Equation (7.1) is small comparedwith unity (paraxial radiation approximation), which is satisfied inpractice for low order modes, then the binomial theorem may be used torewrite Equation (7.1) as: ##EQU4## where a, b, m and n are aspreviously defined, and λ is the free space wavelength of the radiationpropagating in the waveguide.

Equation (6) sets out the electric field contributions obtainable fromall linearly polarized modes of a rectangular waveguide. It iscalculated on the basis that the electric field contribution of eachmode is zero at the side walls of the rectangular waveguide, ie at y=+band -b where y=0 on the equivalent of the axis 18. This is satisfied atleast approximately for a rectangular waveguide with reflecting sidewalls. Not all waveguide modes are necessarily excited by a given input.In the case of the device 10 of FIG. 1, the guide 12 is a special caseof a rectangular guide, since it is of square section. It receives inputof TEM₀₀ form from the apertures 20 and 22. This input excitation iscoupled to the various EH_(mn) modes of the guide 12. The input TEM₀₀field distribution E_(G) say consequently becomes decomposed into alinear combination of the EH_(mn) modes with respective complexmultiplicative coefficients A_(mn). This is expressed by: ##EQU5##

Essentially the A_(mn) amplitude coupling coefficients are thecoefficients of a Fourier series with represents the electric field ateither guide aperture 20 or 22. The EH_(mn) modes are mutuallyorthogonal, and in consequence the coefficients A_(mn) can be calculatedfrom overlap integrals, which, for a rectangular waveguide, are of theform: ##EQU6##

From Equations (7) to (9) it is possible to calculate how the amplitudecoefficients of the excited guide modes vary as a function of w₀ /a, theratio of the beam waist to the aperture size.

Equation (7.2) may be employed to demonstrate modal dispersion within arectangular waveguide, and the consequent field reproduction phenomenonproduced thereby. Putting m=1 and n=p in Equation (7.2) gives the phasecoefficient β_(1p) of guide mode EH_(1p) : ##EQU7## and the phasecoefficient β_(1q) of guide mode EH_(1q) is given by: ##EQU8##

By subtraction of Equation (11) from Equation (10) and rearranging, thephase difference between modes EH_(1p) and EH_(1q) at guide z is X_(z)given by: ##EQU9##

If the condition is imposed that a 2π phase difference is required toexist between the modes, Equation (12) becomes: ##EQU10## and thepropagation distance z (say z₂π) in Equation (13) in rectangularwaveguide that gives rise to a 2π phase difference between modes EH_(1p)and EH_(1q) is given by: ##EQU11##

For the case of the EH₁₁ and EH_(1n) modes (ie the fundamental andn^(th) order odd mode) z₂π is given by: ##EQU12##

Combining Equations (2) and (13): ##EQU13##

With n=3, 5, 7, 9, 11 . . . , z₂π is L', L'/3, L'/6, L'/10, L'/15 . . .. This shows that there is a harmonic relationship between EH_(1n) modesof a rectangular guide. Equation (16) shows that the propagationdistance z₂π which gives rise to a 2π phase shift between thefundamental EH₁₁ mode and the next highest order EH₁₃ mode also givesrise to a 2π phase shift between the fundamental and all other EH_(1n)modes (n odd). This results in reproduction of any symmetric inputelectric field after a distance z₂π, provided that there is noexcitation of even numbered EH_(1n) modes. If there is sufficient lengthof waveguide available, a symmetric input field will be reproducedperiodically at distances of tz₂π, where "t" is an integer number.

It can be shown that similar remarks apply to modes in the orthogonaldimension (mode number m) of a rectangular waveguide; ie if onlysymmetric modes are excited for which m is an odd number, these modeswill be in phase with one another once more (2rπ phase difference, r=0,1, 2, . . . ) at a length of waveguide 4a² /λ, where 2a is the waveguidecross-sectional extent (width or height) in the direction of modesnumber m. The guide 12 is of square section with side 2a (ie a=b) andlength L equal to 4a² /λ. It receives a quasi-TEM₀₀ input at eachaperture 20 or 22 from the respective mirror 14 or 16, and this inputtherefore excites only symmetric modes (m and n both odd). The symmetricmodes which are in phase at one point in a waveguide are also in phaseat positions distant by multiples of L from that point by virtue ofEquation (16). In consequence, provided that departures from Equation7.2 and differences in attenuation of different modes are notsignificant, an electric field distribution input to the guide 12 whichis on axis and located at one of the apertures 20 and 22 will bereproduced at the other of these apertures, provided that only symmetricmodes of the guide are excited.

Referring now to FIG. 3, there is shown a further embodiment of awaveguide laser of the invention indicated generally by 60. The laser 60is equivalent to the FIG. 1 embodiment with one concave mirror 14replaced by a plane mirror 62 placed very close to a guide 64. The guide64 and a concave mirror 66 have identical dimensions and positioningrelative to one another as the like for equivalent elements 12 and 16 inFIG. 1.

The plane mirror 62 is spaced less than 5 mm from the guide 64. Ittherefore falls within the class of Case I mirrors as previouslydefined. The concave mirror 66 is located in a position whichcorresponds (as in FIG. 1) neither to Case I, nor to Case II nor to CaseIII. It is a phase-matched mirror in the medium field.

The laser 60 operates equivalently to that described with reference toFIG. 1. A TEM₀₀ intensity distribution centred on a first guide endaperture 68 diffracts to the phase-matched concave mirror 66 and isreturned to the first aperture 68. The guide 64 reproduces this electricfield distribution at a second end aperture 70 adjacent the plane mirror62, which is for practical purposes at zero separation from the guide.The electric field distribution at the second aperture 70 is accordinglyrecreated at the first aperture 68 and a further optical round tripcommences.

Referring now to FIG. 4, there is shown a further waveguide laser of theinvention indicated generally by 80. This incorporates a fullyreflecting plane mirror 82, a guide 84 and a partially reflectingconcave mirror 86. The laser 80 is exactly as described with referenceto FIG. 4, except that the guide 84 is 2a² /λ in length, half that ofthe earlier equivalent guide 64. In view of the similarity of the lasers60 and 80, only differences in operation will be discussed. The guide 84is of length L/2, using the nomenclature of Equation (1) and FIG. 2. Ittherefore divides a quasi-TEM₀₀ intensity distribution at a first (righthand) end aperture 88 into a four-lobed intensity distribution at asecond end aperture 90. The latter distribution is shown at 44 in FIG.2. By virtue of retroreflection at the plane mirror 82, the four-lobedintensity distribution recreates a single-lobed quasi-TEM₀₀ intensitydistribution at the first aperture 88. Retroflection at the plane mirror82 doubles the effective length of the guide 84 so that a quasi-TEM₀₀intensity distribution is recreated over a path length of L or 4a² /λwithin the guide. If the plane mirror 82 were to be partiallyreflecting, the laser 80 would provide a four-lobed output. Referringnow to FIG. 5, there are shown two graphs 100 and 102 obtained bycalculation and indicating the transmission properties of guides such as12, 64 and 84. The guide cross-section is square of side 2 mm, and theradiation wavelength is 10.59 microns as in the foregoing embodiments.TEM₀₀ transmission fidelity is plotted as a function of length of guide.The expression "transmission fidelity" is defined as the proportion ofinput intensity present in an output TEM₀₀ mode after transmissionthrough a guide of length indicated by the relevant position on thehorizontal axis. The input radiation is taken to have a beam waist w₀ ata guide input aperture, and the output radiation to have a like beamwaist at a guide output aperture. Losses have been estimated based onalumina guide walls. The graphs 100 and 102 assume that the beam waistto guide aperture ratios w₀ /a are 0.3 and 0.5 respectively. The graphs100 and 102 show over 95% transmission fidelity at respective peaks 104and 106 centred at guide length 37.8 cm, which is 4a² /λ or L for 2a=2mm and λ=10.59. This demonstrates the accuracy of reproduction of aTEM₀₀ mode by a guide of length L.

Referring now to FIG. 6, there is graphically illustrated thetheoretically calculated loss as a function of guide length in awaveguide laser resonator experienced by laser radiation in a singleround trip of the laser's internal optical path. In FIG. 1, the roundtrip is a double pass (forward and return) of the laser resonatorbetween the cavity mirrors 14 and 16, and the trip is of length 2(z₁ +z₂+L).

The round trip loss shown in graph 110 relates to the lowest orderresonator mode. The round trip loss shown in graph 112 is that of thesecond order resonator mode.

The graph 110 shows that the round trip loss is low, less than 10%, forthe lowest order resonator mode for wide ranges of values either side ofguide lengths 10.9 cm and 37.8 cm respectively. These guide lengths arethose of the guides 84 and 64 in FIGS. 4 and 3, and correspond to 2a² /λand 4a² /λ respectively. Similar results are obtainable for guidelengths 2na² /λ, where n=3, 4 . . . . The graph 110 shows that, providedthe guide length is a multiple of 2a² /λ, the round trip loss isinsensitive to inaccuracies in guide length. Furthermore, since guidelength is related to a² /λ, minor inaccuracies in guide cross-section donot affect round trip loss. This is a major benefit of the invention,since it ensures that a waveguide laser of the invention having L equalto 2ma² /λ (m=1, 2, 3, . . . ) will produce a substantiallysingle-lobed, on-axis output despite inaccuracies of manufacture withinreasonable tolerances. The relative insensitivity of the invention toguide cross-section errors is particularly important for guides in theregion of 2 mm square, since it is very difficult to maintain accuracyof such a small cross-section over a guide length in the region of 20 or20 cm. For example, a 5% inaccuracy in guide cross-section, ie a 100 μmwidth change, is equivalent to a 10% error (3.8 cm) in guide length byvirtue of Equation (3).

The graphs 110 and 112 are well separated in the regions of guidelengths 18.9 cm and 37.8 cm, the round-trip loss difference being in theregion of 5%. In consequence, for guide lengths within about ±10% of18.9 cm and about ±5% of 37.8 cm, laser action will take placepreferentially in the lower loss fundamental resonator mode to whichgraph 110 relates, and good mode discrimination is obtained. Similarremarks apply with reducing tolerances to guide lengths which are highermultiples of 2a² /λ.

Referring now to FIG. 7, there is shown a schematic drawing of first,second and third waveguide lasers 120a, 120b and 120c (collectivelyreferred to as 120) differing only in guide length. The lasers haverespective guides 121a, 121b and 121c (collectively 121), and respectiveconverging cavity mirrors 122a/123a, 122b/123b and 122c/123c. Lines124a, 124b and 124c indicate outermost radiation intensity contours atwhich intensity is a fraction 1/e² of maximum intensity. The guides 121are of square internal cross-section of side 2a. The first guide 121a isof length 12a² /λ, and the second and third guides 121b and 121c arerespectively shorter and longer than this. Longitudinal positions suchas 125a of maximum separation of intensity contour lines are those atwhich there is division of intensity into a four-lobed pattern in aplane transverse to the length of a guide such as 121a. Longitudinalpositions such as 126a and 127a at which line separation is a minimumare those at which a laser beam waist occurs; ie the transverseintensity pattern is single-lobed and on the waveguide axis.

Although it is a schematic drawing, and length differences between theguides 120 have been exaggerates for reasons of clarity, FIG. 7indicates that guide length can vary without greatly affecting theradiation intensity distributions within the laser cavities defined bymirror pairs 122a/123a etc.

However, outermost beam waist positions 126b and 126c become outside andinside respective guides 121b and 121c as a result of change of guidelength. This illustrates that inaccuracy in guide length shifts beamwaist position from the design location in the plane of a guide aperture(eg aperture 20 or 22 in FIG. 1). It also alters the beam waist sizesomewhat. Because guide length is proportional to the square of theguide cross-sectional width by Equation (18), inaccuracy in guidecross-section has the same effect of shifting beam waist position. Inconsequence, it is only possible to define a design position and sizefor a beam waist, since manufacturing errors will change the beam waistposition and size from their design values. Similar remarks apply toerrors in mirror positions.

Referring now to FIG. 8, there is shown a further embodiment of awaveguide laser of the invention indicated generally by 140. Itincorporates first and second guides 142 and 144 together with first,second and third concave mirrors, 146, 148 and 150. The laser 140 isequivalent to two lasers 10 superimposed and coupled together, with atilted second mirror 148 reflecting radiation from one guide 142 or 144to the other 144 or 142. One of the mirrors 146, 148 and 150 ispartially reflecting and the other two are fully reflecting. If one ofthe first and third mirrors is partially reflecting, a laser output beamemerges from it. If the second mirror 148 is partially reflecting, thereare two mutually inclined output beams each coaxial with a respectiveguide and each phase-locked to the other beam.

FIG. 9 shows a further embodiment of the invention indicated generallyby 160. It incorporates a concave mirror 162, a guide 164, a lens 166and a diffraction grating 168 inclined at the Littrow angle θ to a laserbeam 170. θis given by:

    θ=sin.sup.-1 (λ/2d)                           (17)

where λ is the laser wavelength and d is the grating line spacing.

The combination of the lens 166 and grating 168 acts as a substantiallyphase matched, retroreflecting mirror equivalent to the mirror 16 inFIG. 1. Phase matching is not exact for reasons previously given andbecause of the grating inclination to the beam 170. Lens-gratingcombinations equivalent to mirrors are well-known in the art of leasersand will not be described further.

FIG. 10 shows an embodiment of the invention indicated generally by 180and equivalent to that described with reference to FIG. 8. Thedifference between these embodiments is that the tilted second mirror148 in the latter has been replaced by two lenses 182, 184 and a tiltedplane mirror 186 in the former. The embodiment shown in FIG. 10 hasfirst and second concave mirrors 188 and 190 and first and secondwaveguides 192 and 194 identical to equivalents shown in FIG. 8. This isan example of a lens and plane mirror combination being equivalent to aconcave mirror. A further alternative is a combination of a lens andcurved mirror.

The guides 12, 64, 84, 142, 144, 164, 192 and 194 all have squarecross-sections. It is also possible to employ a rectangularcross-section guide having sides 2a by 2b (b>a). In this case, to obtainelectric field preservation, the relationship between guide length L,guide width and wavelength λ within the guide is to be satisfied forboth parameters b and a simultaneously. In consequence, for arectangular section guide of internal dimensions 2a×2b×L: ##EQU14##

Equation 19 shows that a rectangular guide obeying Equation 18 will beelectric field preserving if its transverse dimensions 2b and 2a have aration which is the square root of an integer ratio. The laser beamwaist radius w₀ for a laser employing such a waveguide is required to bein the range 0.1b to 0.65b in the width dimension of side 2b and 0.1a to0.65a in the width dimension of side 2a.

In a laser analogous to the FIG. 4 device 80, equivalents of Equations(18) and (19) are as follows:

    L=2mb.sup.2 /λ=2na.sup.2 /λ                  (20)

and

    b/a=√(n/m) as before (                              (21)

It is also possible to employ a one dimensional guide in a laser of theinvention. Such a guide has two substantially planar walls which aresubstantially parallel to one another. The walls provide waveguidingwith respect to one transverse dimension. There are no guide walls orwaveguiding effects in the orthogonal transverse dimension; in thislatter dimension, the laser acts as a conventional free space resonatorhaving reflecting means and a gain medium but no guide. The magnitude ofthe laser beam waist (2w₀) should be construed as a minimum beam width,sine in this example the intensity distribution at the beam waist is notsymmetric with respect to rotation about the laser optical axis.

As a further example, a laser of the invention may be configured inaccordance with the invention in one transverse dimension but not in theother. In this example, the mirror and guide geometry in one transversedimension is as in the preceding one dimensional case. In the othertransverse dimension, the resonator geometry may be as in any prior artlaser arrangement; eg it may include guide walls not positionedequivalently to those of the one dimensional case.

We claim:
 1. A waveguide laser including a waveguide located in a laserresonator cavity defined by first and second reflecting means, andwherein:(a) the waveguide has at least one pair of substantially planarguide walls which are substantially parallel to one another, andseparated from one another by a distance 2a; (b) the cavity is designedto produce a beam waist of magnitude of 2w₀ located centrally of awaveguide end aperture, where w₀ is within a range of 0.1a and 0.65a;(c) the first reflecting means is located to receive radiation emergentfrom the waveguide through the end aperture, and has converging andreflecting properties which, at least in a dimension orthogonal to theguide walls, are arranged to be phase matched to radiation received froma TEM₀₀ amplitude distribution at the aperture and having said beamwaist magnitude; and (d) the cavity is designed to be electric fieldpreserving at the waveguide end aperture with a TEM₀₀ radiationamplitude distribution at this aperture and having said beam waistmagnitude is designed to be recreated after radiation therefrom haspassed through the waveguide to the second reflecting means andreturned.
 2. A laser according to claim 1 wherein w₀ is in a range 0.3ato 0.5a.
 3. A laser according to claim 2 wherein the waveguide is afirst waveguide and the laser includes a second waveguide within thecavity.
 4. A laser according to claim 3 including means for couplingradiation from the first waveguide to the second waveguide which meansdefines mutually inclined optical paths in the first and secondwaveguides.
 5. A laser according to claim 1 having a gain medium withinthe waveguide providing gain at an operating wavelength within thewaveguide of λ, and wherein the waveguide is of square cross-sectionwith side 2a and length 4na² /λ where n is a positive integer, the endaperture is a first such aperture and the waveguide has a second endaperture at which the cavity is arranged to be electric fieldpreserving.
 6. A laser according to claim 1 having a gain medium withinthe waveguide providing gain at an operating wavelength within thewaveguide of λ, and wherein the second reflecting means is a planemirror immediately adjacent the waveguide, the waveguide is of squarecross-section with side 2a and length 2a² /λ, the aperture is a firstend aperture and the waveguide has a second end aperture arrangedimmediately adjacent the second reflecting means.
 7. A laser accordingto claim 1 having an operating wavelength within the waveguide of λ, andwherein:(a) the waveguide is a first waveguide, (b) the aperture is oneof two end apertures of the first waveguide, (c) a second waveguidehaving two end apertures is arranged within the cavity, (d) the laserincludes means for coupling radiation between one end aperture of thefirst waveguide and one end aperture of the second waveguide, and (e)both waveguides are of square cross-section with side 2a and length 4a²/λ.
 8. A laser according to claim 7 wherein the means for couplingradiation defines mutually inclined optical paths in the first andsecond waveguides.
 9. A laser according to claim 1 wherein at least oneof the first and second reflecting means comprises a leans arranged incombination with either a mirror or a diffraction grating.
 10. Awaveguide laser including a waveguide located in a laser resonatorcavity defined by first and second reflecting means, said waveguideincluding at least one waveguide end aperture, and wherein:(a) thewaveguide has at least one pair of substantially planar guide wallswhich are substantially parallel to one another, said guide walls havinga separation distance equal to 2a, where "a" is equal to one half ofsaid separation distance; (b) the cavity comprises a means for producinga laser beam waist located centrally of said at least one waveguide endaperture, said beam waist having a magnitude of 2w₀, where w₀ is equalto one half of said magnitude and is within a range of 0.1a and 0.65a;(c) the first reflecting means is located to receive radiation emergentfrom the waveguide through said at least one waveguide end aperture, andis phase matched to radiation arising from laser action within thecavity, at least in a dimension orthogonal to the guide walls, and saidlaser radiation having a TEM₀₀ amplitude distribution at the apertureform and having said waist magnitude of 2w₀ ; and (d) the cavity and thewaveguide in combination comprising a means for electric fieldpreservation at said at least one waveguide end aperture wherein thelaser produces a laser radiation having a TEM₀₀ amplitude distributionand having said beam waist magnitude of 2w₀, and said amplitudedistribution becomes recreated by radiation therefrom passing throughthe waveguide to the second reflecting means and returning.
 11. A laseraccording to claim 10 wherein said parameter w₀ is within a range of0.3a to 0.5a.
 12. A laser according to claim 11 wherein the waveguide isa first waveguide and the laser includes a second waveguide within thecavity.
 13. A laser according to claim 12 further including means forcoupling radiation from the first waveguide to the second waveguidewherein said means for coupling defines mutually inclined optical pathsin the first and second waveguides.
 14. A laser according to claim 11having a gain medium within the waveguide providing gain at an operatingwavelength of λ within the waveguide, and said waveguide is of squarecross-section with 4 sides, each side having a width equal to 2a and alength equal to 4na² /λ where n is a positive integer, the at least oneend aperture is a first such aperture and the waveguide has a second endaperture, and the cavity and the waveguide in combination comprising ameans for electric filed preservation at the second end aperture.
 15. Alaser according to claim 10 having a gain medium within the waveguideproviding gain at an operating wavelength of λ within the waveguide, andsaid waveguide is of square cross-section with 4 sides, each side havinga width equal to 2a and a length equal to 4na² /λ where n is a positiveinteger, the at least one end aperture is a first such aperture and thewaveguide has a second end aperture, and the cavity and the waveguide incombination comprising a means for electric field preservation at thesecond end aperture.
 16. A laser according to claim 10 having a gainmedium within the waveguide providing gain at an operating wavelength ofλ within the waveguide, and wherein the second reflecting means is aplane mirror immediately adjacent the waveguide, the waveguide is ofsquare cross-section with 4 sides, each side having a width equal to 2aand a length equal to 2a² /λ, the at least one aperture is a first endaperture and the waveguide has a second end aperture arrangedimmediately adjacent the second reflecting means.
 17. A laser accordingto claim 10 having an operating wavelength of λ within the waveguide,and wherein:(a) the waveguide is a first waveguide, (b) said at leastone waveguide end aperture is one of two waveguide end apertures in thefirst waveguide, (c) a second waveguide having two waveguide endapertures is arranged within the cavity, (d) the laser includes meansfor coupling radiation between one waveguide end aperture of the firstwaveguide and one waveguide end aperture of the second waveguide, and(e) both waveguides are of square cross-section with 4 sides, each sidehaving a width of 2a and a length of 4a² /λ.
 18. A laser according toclaim 15 wherein the means for coupling radiation defines mutuallyinclined optical paths in the first and second waveguides.
 19. A laseraccording to claim 10 wherein at least one of the first reflecting meansand second reflecting means comprises a lens arranged in combinationwith one of a mirror and a diffraction grating.